University of LjubljanaFaculty of Mathematics and Physics

Department of Physics

Seminar Ib – 1st year, 2nd cycle

The cesium frequency standard

Author: Jan Fiser

Advisor: assoc. prof. Igor Poberaj

Ljubljana, June 2014

Abstract

In the seminar the cesium atomic clock, which serves as the primary frequency standard usedfor the definition of the second, is described. First, a brief history of time measurement andaccompanying clock development is described, followed by general considerations about clockdesign and the definition of Allan variance as a measure of frequency stability. Then, thedefinition of the second and its history are given. The main part of the seminar consists ofthe description of classical, laser pumped and fountain cesium standards, along with sourcesof systematic errors. At the end, the future prospect of the time standard is discussed andapplications based on accurate time measurement are presented.

Contents

1 Introduction 3

2 A brief history of time measurement 3

3 General clock design 4

4 Stability 4

5 The definition of the second and its history 5

6 The classical Cs beam standard 56.1 The classical Cs beam standard scheme . . . . . . . . . . . . . . . . . . . . . . . 56.2 Components of the classical Cs beam standard . . . . . . . . . . . . . . . . . . . 6

7 The laser-pumped cesium beam standard 7

8 The cesium fountain standard 88.1 Laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

9 Sources of systematic errors 10

10 Future development of the time standard 11

11 Applications based on accurate time measurement 11

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1 Introduction

Time has always played an important role in human lives. Surprisingly, people are often unawareof how exactly time is measured and its unit defined. The aim of this seminar is to dispel thisignorance.

A brief history of time measurement and accompanying clock development is described inSection 2, where we closely follow [1]. A lesson to be learnt from it is that the universal accessi-bility of time, nowadays, is the consequence of centuries-long struggles. General considerationsabout clock design and performance are derived in Sections 3 and 4 based on historical examples.Then, the definition of the second and its historical development is outlined in Section 5. Fromthe formal definition of the second it is hard to imagine how it is measured in practice. Thisis clarified in Sections 6, 7 and 8, where different experimental apparatus are described, whileSection 9 focuses on their systematic uncertainties. The future prospect of time standards isdiscussed in Section 10. The concluding Section 11 lists some applications of commercial andstate-of-the-art atomic clocks.

2 A brief history of time measurement

The ancient civilizations recognized the cycles of the Sun, the Moon and the stars. Observationand measurement of their periodic motions led to the natural time units of days, months andyears. Such time division sufficed for daily life, religious practices and agriculture, so it formedthe basis for timekeeping.

The division of day into smaller units stimulated the development of man-made clocks. Oneof the first was the shadow clock, which was later refined into the sundial. Beside clocks basedon heavenly bodies, other types of clocks were developed, e.g. water and sand clocks. Theaccuracy of all these clocks was approximately equal.

In the beginning of the 14th century weight-driven mechanical clocks were developed inEurope. However, they were highly inaccurate, with time discrepancies of about 15 minutes perday. A breakthrough occurred when Christian Huygens built the first pendulum-based clock in1656 with an accuracy of 10 seconds per day.

A strong motivation for the development of a reliable and robust clock was the need forprecise navigation (determining longitude) at sea. Pendulum clocks were obviously unsuitable,so mechanical clocks based on springs were developed. John Harrison built a clock in 1759 whichlost only 5 seconds in 81 days at sea. The importance of accurate clocks further increased withthe development of railroad transportation.

Throughout the 18th and 19th century mechanical clocks were steadily improved. In 1921William Hamilton Shortt built a two-pendulum clock with an accuracy of a millisecond per day.It consisted of a master pendulum, which kept the time, and a slave pendulum, which providedimpulses to the master (to compensate energy losses). The two pendulums were electricallycoupled and housed in vacuum.

The complexity of Shortt’s clock indicated that mechanical clocks had more or less reachedtheir highest attainable accuracy. A new approach was needed for any further significant im-provement of the accuracy. A breakthrough occurred in 1927 when Marrison and Horton builtthe first quartz clock. The principle of operation is as follows: an electronic circuit forces a tinytuning fork made of quartz crystal to vibrate at its resonant frequency via the inverse piezoelec-tric effect. Consequently, the tuning fork generates an electric signal at its resonant frequencyvia the direct piezoelectric effect. Another circuit detects the signal produced by the quartzoscillator and lowers its frequency to 1 Hz. These regular 1 Hz pulses power a stepping motorthat turns the clock’s hands. With such design, many problems associated with pendulum clocksare greatly reduced, so that the accuracy of a good quartz clock is around one millisecond peryear. The accuracy is limited by temperature variations, which cause slight changes in size of

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the crystal and subsequent drifts of its resonant frequency.Even greater accuracy is achieved by atomic clocks, which will be described in detail later.

3 General clock design

Roughly speaking, every clock (excluding ancient clocks) has three essential functional parts: anenergy source, a system capable of stable periodic motion to serve as regulator and a mechanismto derive and display the time in the desired units [2].

The energy source is typically a falling weight, a coiled spring, a battery, or some otherelectric source. A mechanism to derive and display the time is usually a system of gear trainsand a dial or an electronic display.

The most critical is the regulating system. The ideal situation would be the one in whichthe regulator would oscillate freely at its resonant frequency, regardless of external conditions.Furthermore, the control of energy transmission from the energy source to the rest of the clockmechanism would leave the regulator unperturbed. However, this is never realised in practice.Firstly, the oscillator’s period is always a function of environmental conditions. But with carefuldesign we can control them and calculate their influence on the oscillator’s period. Secondly,energy losses are always present in the regulator, so a small driving force must act on it tomaintain constant oscillations. This contradicts the requirement of an unperturbed regulator.To accommodate these opposing requirements, the oscillator must have very small inherentenergy losses, so that only a very weak driving force is needed [2].

An ideal oscillator would be one that, given a single initial push, would run forever. Asstated above, this is never realised in practice because of energy losses. The quality factor Q isintroduced as the ratio of the total energy in a system to the energy lost per cycle. If we restatethis in resonance-curve language, the quality factor is expressed as

Q =ωres

∆ω,

where ωres is the resonant frequency and ∆ω is the bandwidth (full width at half maximum) ofthe curve [1]. Oscillators with high Q are obviously more suitable for regulators, because theyneed weaker driving forces and their oscillation frequency is more sharply determined.

4 Stability

The stability of frequency (time) standard is obviously of fundamental importance. The standardis subject to fluctuations and long-term drifts. The former are assumed to be random andstationary, while the latter are deterministic. In time-domain description, errors in frequencyover different sampling intervals (sources of instability may have different time dependence) oftime are statistically analysed [2]. The accepted measure of instability is Allan variance, definedby [1]

σ2 =1

2(N − 1)

N−1∑i=1

(ni+1 − ni)2,

where ni denotes frequency count on the i-th measurement of the same sample interval, which ismeasured N times. By subtracting the previous frequency count ni (instead of the mean) fromeach frequency count ni+1 in the summation, Allan variance eliminates any systematic errorcontribution.

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5 The definition of the second and its history

The concept of an hour being divided into 60 minutes, each in turn composed of 60 seconds, wasalready present in the middle of the 14th century (the base 60 was probably chosen because it isdividable by: 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30). However, the concept did not find widespreadusage because clocks were not capable of measuring minutes and seconds reliably. Clocks withsecond hands did not appear until the 18th century [1].

The first formal definition of the second was proposed by Tito Livio Burattini in 1675 as1/86 400 of a mean solar day.

The definition of the second based on the Earth’s variable rotation became impractical forprecise timekeeping in the 20th century. In 1960 a new definition of the second was establishedas the fraction 1/31 556 925.9747 of the length of the tropical year for 1900. This was called theEphemeris second.

The development of the cesium (Cs) frequency standard took place at the same time as thediscussion and adoption of the Ephemeris second. The first Cs beam atomic clock was built byEssen and Parry in 1955. The frequency of the Cs standard had to be calibrated against theEphemeris second. This was done by Markowitz in 1958 [3]. Soon after the adoption of theEphemeris second, it became evident that the atomic standard of time could be realized andreproduced more accurately. In 1967 the SI definition of the second was established: The secondis the duration of 9 192 631 770 periods of the radiation corresponding to the transition betweenthe two hyperfine levels of the ground state of the cesium 133 atom. This definition refers to aCs atom at rest at a temperature of 0 K [1]. The definition of the second should be understoodas the definition of the unit of proper time: it applies in a small spatial domain which shares themotion of the caesium atom used to realize the definition. The proper second is obtained afterapplication of the special relativistic correction for the velocity of the atom in the laboratory [4].

6 The classical Cs beam standard

First, a rough description of the classical Cs beam standard is given. After the principle ofoperation is clarified, each component is described in detail.

6.1 The classical Cs beam standard scheme

The first operational Cs beam atomic clock was built in 1955 by Essen and Parry [1]. Its schemeis shown in Fig. 1.

Figure 1: A beam of Cs atoms leaves the oven. A state-selective magnet deflects atom’strajectory according to their quantum state. Atoms in the chosen group of states pass througha region of a weak uniform magnetic field and are irradiated by microwaves. If the frequency ofthe microwaves is appropriate, transitions between the chosen quantum states occur. Anotherstate-selective magnet acts as an analyser, separating atoms that made the transition from therest and directing them at the detector. (Figure taken from [1].)

Cs vapour is heated in the oven to T ≈ 100oC. Cs atoms leave the oven through the narrow

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slit, forming a beam. The A-magnet produces strong nonuniform magnetic field, which acts asa state selector, because Cs atoms are deflected according to their quantum state. The chosengroup of states passes to the region of a weak uniform magnetic field, called the C-field. Ontheir way through the C-field area, atoms pass through a microwave cavity. If the frequencyof the microwave field is appropriate, transitions between chosen quantum states occur. Theatoms pass through another state-selective magnet, called the B-magnet. Those that made thetransition are deflected away from the rest of the Cs beam in the direction of the detector. Thenumber of atoms that hit the detector is converted into an electric signal. By adjusting themicrowave frequency, the electric current is maximized. At its maximum, the frequency of themicrowave field is equal to the “frequency between chosen quantum states”. This frequency isused in the SI definition of the second.

6.2 Components of the classical Cs beam standard

In this subsection components of the classical Cs beam standard are described in detail. Furtherexplanations are found in [2].

Cs is heated in the oven to around 100oC. The Cs vapour passes through the collimator, sothat the emerging beam of Cs atoms is as narrow as possible. Higher temperatures are not used,because collisions between atoms would be more probable, which could lead to Cs build-up inthe collimator and causing fluctuations in the beam intensity.

Knowing the detailed structure of the Cs energy spectrum is crucial for understanding theeffect of state-selective magnets A and B. We limit ourselves to listing the facts, which areexplained in [5].

The ground state electronic configuration of cesium is [Cs] = [Xe] 6s1. Completely filledelectronic shells have no angular momentum. Orbital angular momentum L and spin angularmomentum S of the outermost electron are L = 0 and S = 1

2 , hence its total angular momentumJ = L + S has magnitude J = 1

2 . The Cs nucleus also possesses angular momentum I withmagnitude I = 7

2 . The total angular momentum of the Cs atom F = J + I can thus have amagnitude of either F = 3 or F = 4. Nuclear and electronic angular momenta are coupled byhyperfine interaction (Hhf ∝ J · I). Consequently, states with lower F have lower energy. Inexternal magnetic field B, states are split according to their magnetic quantum number mF asshown in Fig. 2.

Figure 2: External magnetic field lifts the degeneracy of the hyperfine levels. Each hyperfinelevel fans out into 2F + 1 magnetic sublevels, characterized by mF . (Figure taken from [5].)

The energy shift due to the applied magnetic field represents a potential energy, whosegradient determines the magnetic force on the atom. In the limit of strong magnetic field, twogroups of states are formed (Fig. 2). These two groups are separated by the A-magnet. Thegroup containing |F = 3,mF = 0〉 is led further into the apparatus, while the group containing|F = 4,mF = 0〉 is absorbed on the getters.

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The Cs beam then enters a magnetically shielded area, where transitions between |F =3,mF = 0〉 and |F = 4,mF = 0〉 are induced. These hyperfine states are chosen because theyare the least magnetic-field-sensitive sublevels. Inside the shielded area, a weak magnetic field,called the C-field, is produced. The C-field has to be weak, so that mF = 0 states do notexperience significant energy shifts (Fig. 2). On the other hand, it has to be strong enoughto cause sufficient energy separation between mF = 0 and mF 6= 0 states, so that transitionsbetween states |F = 3,mF = 0〉 and |F = 4,mF 6= 0〉 are not induced as well.

Transitions between |F = 3,mF = 0〉 and |F = 4,mF = 0〉 are induced with microwaves. Inprinciple, when energy and polarization of microwave photons are appropriate, the transition islikely to occur. Achieving this in practice is a difficult task, plagued with many technical details,which can be found in [2, 6, 7].

The purpose of the B-magnet is to separate the atoms that made the transition from therest. The principle of operation is the same as in the case of the A-magnet.

The detector is a “hot-wire detector” based on surface ionization of Cs atoms. The impingingCs atoms give their outermost electrons to the material of the wire and leave as positively chargedions. This process is possible when the metal’s work function is greater than the binding energy ofthe outer electron in Cs (≈ 3.87 eV). The wire is made of certain pure metal, which has to be hotto prevent surface adsorption of gases. The materials in use are tungsten, niobium, molybdenumand Pt-Ir alloy [2]. The Cs ions are gathered by the cathode of an electron multiplier, whichproduces a measurable electric signal, proportional to the number of incident Cs ions. Theseare again proportional to the number of Cs atoms that made the transition. Maximizing theelectric signal means that the frequency (and thus energy) of microwaves matches the frequency(energy) difference between chosen hyperfine states.

However, the Cs beam resonator is a passive device, which merely serves as a frequencyreference. We chose that when electric signal from Cs clock is at its maximum, the frequencyof microwaves is by definition 9 192 631 770 Hz. For a functional clock, a time display and anelectronic circuit, which locks the tunable oscillator’s frequency on the Cs resonance frequency,is needed. The latter is achieved with a phase-locked loop. The conceptual scheme of the atomicclock is shown in Fig. 3. The actual clock’s size depends on its purpose – primary standardclocks are the size of a small car, but commercial (and less accurate) clock are typically the sizeof a suitcase.

Figure 3: The conceptual scheme of the atomic clock. A tunable oscillator generates thefrequency of the microwaves, which is locked to the Cs resonant frequency with a phase-lockedloop. The atomic resonator serves merely as a frequency reference. The tunable oscillatorgenerates another signal, whose frequency is stepped down to 1 Hz, used for controlling the timedisplay. (Figure taken from [1].)

7 The laser-pumped cesium beam standard

The laser-pumped Cs beam standard differs from the classical Cs beam standard in the aspectof state selection and the detection of clock transitions. State-selective magnets are replaced

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by optical pumping and hot-wire detector by fluorescence detector. Otherwise everything is leftunchanged, so the overall complexity of the clock remains approximately the same as in the caseof classical Cs beam standard. The first optically pumped Cs clock was made in 1980 by Arditiand Picque [1].

By using the A-magnet, the blue branch of states in Fig. 2 is discarded. This results insubstantial signal-to-noise (SNR) decrease. The main idea of optical pumping is to increase thedifference between |L = 0, F = 3〉 and |L = 0, F = 4〉 states before Cs atoms enter the microwavecavity. This can be achieved by many different schemes. The following example highlights theconcept of optical pumping.

When Cs atoms leave the oven, the population of |L = 0, F = 3〉 and |L = 0, F = 4〉 levels areapproximately the same [8]. The transition |L = 0, F = 4〉 → |L = 1, F = 4〉 is excited with anappropriate laser (Fig. 4). The excited state decays back to |L = 0, F = 3〉 and |L = 0, F = 4〉with a certain branching ratio. The result is a net transfer from |L = 0, F = 4〉 to |L = 0, F = 3〉state, so more Cs atoms are “available” to make the resonant transition and therefore SNR isincreased.

Figure 4: The concept of optical pumping. Initially, both quantum states |L = 0, F = 3〉and |L = 0, F = 4〉 are equally populated. Transitions from |L = 0, F = 4〉 to |L = 1, F = 4〉are excited with a laser. The excited state decays back to the lower-lying states with a certainbranching ratio, resulting in a net transfer from |L = 0, F = 4〉 to |L = 0, F = 3〉. Therefore,more Cs atoms are “available” to make the resonant transition. (Figure taken from [9].)

The optical detection scheme is based on fluorescence. Instead of the B-magnet, a laser istuned to one of the cyclic (closed) transitions from |L = 0, F = 4〉 state, i.e. a transition whenatom after the excitation falls back to its original |L = 0, F = 4〉 state because of selectionrules [10]. The emitted fluorescent light is proportional to the number of atoms that made thetransition to |L = 0, F = 4〉 state in the microwave cavity. Cyclic transition enables nearly 100 %fluorescence detection efficiency since every atom can be analysed as long as it is illuminated bythe detecting laser.

Beside superior SNR and detecting efficiency, the trajectories of atoms do not depend ontheir thermal velocities as they do in the case of the classical Cs beam standard.

8 The cesium fountain standard

Further improvement in accuracy of the Cs standard is achieved by the Cs fountain standard.It differs drastically from both Cs beam standards. The concept was first proposed by JerroldZacharias in 1953 [11]. However, lacking suitable technology, the first operational Cs fountainstandard was not built until 1994 [2].

A group of approximately 107 Cs atoms is laser trapped in 3D space and cooled by threemutually orthogonal laser pairs (Fig. 5). The group of atoms is vertically launched with velocityv ≈ 4 m/s. This is accomplished by introducing a change in the frequency of the vertical pairof lasers. After launching, the lasers are shuttered and atoms move only under the influence of

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gravity. Atoms are optically pumped to the |L = 0, F = 3,mF = 0〉 state on their way towardthe microwave cavity, where they are irradiated by microwaves. Gravity decelerates the atomsand eventually causes them to fall downwards. On their way down, the atoms are irradiated forthe second time in the same microwave cavity. The number of atoms that make the transitionis detected optically via fluorescence. This cycle is repeated when another group of atoms isprepared.

Figure 5: The scheme of the Cs fountain standard. Three mutually orthogonal laser pairstrap and cool a group of Cs atoms. The group of atoms is launched upwards by introducing achange in the frequency of the vertical pair of lasers. The lasers are then shuttered and atomsmove only under the influence of gravity. Atoms are then optically pumped and irradiated bythe microwaves. Gravity decelerates the atoms and causes them to fall downwards. The samemicrowave field irradiates the atoms for the second time. Finally, the number of atoms thatmake the transition is detected optically via fluorescence. (Figure taken from [14].)

As it turns out, the time atoms need to travel from the first to the second microwave cavity,∆t, determines the width of the observed resonance – longer ∆t produce sharper resonances [2].In the classical and the laser-pumped Cs beam standard ∆t ≈ 10 ms. It is determined by themean thermal velocity of Cs atoms vt, which is vt ≈ 250 m/s at 100oC, and the tube length L,which is L ≈ 1 m, limited by the beam deflection due to gravity. The corresponding frequencywidth of the observed resonance peak is ≈ 100 Hz, so the quality factor is ≈ 108. In the Csfountain standard, ∆t ≈ 1 s, because Cs atoms are laser-cooled to temperatures on the orderof µK and launched from a laser trap with velocity v ≈ 5 m/s [8]. In such low-velocity regime,gravity effects become dominant. Instead of the horizontal path of Cs atoms, a vertical pathhas to be chosen. The width of the observed resonance peak is ≈ 1 Hz and the quality factor≈ 1010.

8.1 Laser cooling

Laser cooling techniques were developed in the 1970s [1]. We shall briefly explain only theirprinciple. Further explanations can be found in [2, 12, 13].

Two counter-propagating laser beams with equal intensities and high spectral purity aredirected at an atom (Fig. 6). The laser frequency is tuned slightly below the transition frequencybetween its ground state and some excited state (in practice the excited state is determined bythe availability of lasers with suitable wavelength). The atom is moving, so in its reference frame,the two laser beams frequencies are Doppler-shifted by the same amount, but in the opposite

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direction. The laser beam, whose direction of propagation is anti-parallel to the atom’s velocity,is shifted toward the resonant frequency and vice-versa. Consequently, the atom absorbs morephotons from the direction of its motion and because of momentum exchange, its motion isslowed down.

Figure 6: The principle of so-called Doppler laser cooling in one dimension. Two counter-propagating laser beams with equal intensities are directed at an atom. Their frequency istuned slightly below the transition frequency between the ground state and some excited state.In the reference frame of the moving atom, the frequency is Doppler-shifted. Consequently,the atom absorbs more photons from the direction of its motion and because of momentumexchange, its motion is slowed down. (Figure taken from [13].)

Temperatures on the order of 100 µK can be reached by the described method. For furthercooling, more sophisticated methods have to be used. They are described in [2, 12, 13].

9 Sources of systematic errors

Systematic errors are persistent errors that affect the observed Cs resonance in a deterministicway. They are not caused by unpredictable random fluctuations. The Cs standard has manysources of systematic errors, but that does not mean it is unsuitable for the definition of thesecond, since sources are well understood and their effect calculable [2]. We limit our discussionto a few of them. More on this topic can be found in [9, 10, 15].

Even in weak magnetic fields, the states |F = 3,mF = 0〉 and |F = 4,mF = 0〉 are notcompletely field-independent. The energy shift is quadratic in B (Breit-Rabi formula). Theassociated frequency shift, divided by the resonance frequency, δω

ωres, is on the order of 10−10

[10]. This is the most important frequency shift.In the reference frame of moving Cs atoms the frequency of microwaves is shifted due to

the Doppler effect. The first-order (classical) Doppler effect disappears when the atom movesperpendicularly to the electromagnetic propagation (which is the case in all Cs standards).Since ultra high precision is required of the standard, the relativistic Doppler effect also hasto be taken into the account. The second-order Doppler effect shift is negligibly small in thefountain standard. In the classical standard it causes δω

ωres≈ 10−13 [9, 10].

Atoms interact with ambient thermal blackbody radiation. At T ≈ 300 K the shift isδωωres≈ 10−14 [10].

According to the general theory of relativity clock rate is a function of the gravitationalpotential at the location of the clock. Therefore, the Cs standard frequency is a function of itsaltitude relative to the geoid. The shift is δω

ωres≈ 10−16 m−1 [10].

Asymmetric imperfections in the fabrication of the microwave cavity also cause a frequencyshift. This shift is evaluated by reversing the atom beam direction. The fountain standard usesonly one cavity, so this problem does not arise at all.

Spatial variation in the magnetic field is seen by moving the atoms as an oscillatory field.It could cause transitions between mF sublevels of the same F , known as Majorana transitions.Again, this is problematic only in magnetic state selection scheme [10].

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The relative frequency uncertainty, ∆ωωres

, of the classical Cs beam standard is on the order of

10−13 [16], while the laser-pumped Cs beam standards reach the order of 10−15 [17]. The currentvalue of the fountain Cs standard is on the order of 10−16 [18]. It is limited by the statisticalmeasurement uncertainty.

10 Future development of the time standard

The reasons for choosing Cs for a frequency standard are historical. Firstly, Cs is a hydrogen-likeatom with a single outer electron, so its spectrum is fairly simple. Its fundamental state hasonly two hyperfine levels and at operating temperatures all Cs atoms are in that fundamentalstate. Secondly, the |L = 0, F = 4〉 state has long decay time compared to the observation time.Thirdly, the transition frequency is in the microwave range, which was easily detectable withthe equipment in the 1950s [1].

However, the next generation of frequency standards will probably be based on optical transi-tions. Since optical frequencies are approximately 105 times higher than microwave frequencies,the relative frequency error, ∆ω

ωres, is lower (the order of 10−18 has already been reached [19]),

so they are inherently more suitable. Unfortunately, no electronics exists that is fast enough todirectly count at optical frequencies. Therefore, a lot of research was devoted to this problemin the past two decades. The problem was overcome with the development of frequency combs,which link optical frequencies to lower (microwave) frequencies. There are two major directionsof research: a single ion isolated in an ion trap and many neutral atoms trapped in an opticallattice. Both have already surpassed the performance of the Cs standard, but there is still plentyof research to be done, before they are adopted as the new frequency standard.

11 Applications based on accurate time measurement

There are many everyday applications which rely heavily on accurate time measurement: satellitenavigation, synchronization of telecommunications networks, distribution of electrical power andtime stamping of financial transactions, to name just a few examples [1]. Miniaturization ofatomic clocks led to the development of commercially available chip-scale atomic clocks. Theirlesser accuracy ( ∆ω

ωres∼ 10−11) is compensated by low power consumption (≈ 100 mW), small

volume (≈ 20 cm3) and weight (≈ 40 g) [20]. Currently their price is around $1500, but whenit drops, chip-scale atomic clocks will probably become part of everyday devices.

The above examples require time accuracy on the order of nanoseconds, which is easilyaccommodated with commercial atomic clocks. An application that requires today’s state-of-the-art time accuracy is the measurement of time dilatation predicted by both Einstein’s theoriesof relativity. The ticking rate of two clocks depends on their relative motion and gravitationalpotential they feel. By using two aluminium optical clocks (one clock served as a reference),which keep time to within 1 s/3.7 billion years [21], both effects were measured. The clocks’relative velocity was 10 m/s and altitude difference merely 33 cm [22]. Previous experiments ofthis sort had to be carried out in space, because of lower accuracy of Cs clocks.

Many unifying theories (e.g. string theory) suggest possible variations of the dimensionlessfundamental physical constants such as the fine structure constant α. Atomic transitions indifferent atoms depend differently on α. The explanation of dependence is beyond the scopeof this seminar, but can be found in [23]. A time variation of α induces a change in thefrequency of one clock relative to the other. Comparing the rates of different atomic clocksover several years allows us to put bounds on the local change of α with time. The fractionalvariation of the fine structure constant

∣∣ αα

∣∣ was found to be below 10−17yr−1 [24]. If variationsof fundamental constants are confirmed, Einstein’s equivalence principle (and with it generalrelativity) is violated.

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References

[1] D. D. McCarthy and K. P. Seidelmann, TIME – From Earth Rotation to Atomic Physics(Wiley - VCH, Weinheim, 2009)

[2] F. G. Major, The Quantum Beat: Principles and Applications of Atomic Clocks (Springer,New York, 2007)

[3] W. Markowitz, R. Glenn Hall, L. Essen and J. V. L. Parry, Phys. Rev. Lett. 1, 105 (1958)

[4] SI Brochure Appendix 2: Practical realization of the definition of the unit of time, availableonline at http://www.bipm.org/utils/en/pdf/SIApp2 s en.pdf

[5] D. A. Steck, “Cesium D Line Data”, available online at http://steck.us/alkalidata

(revision 2.1.4, 23 December 2010)

[6] N. F. Ramsey, Journal of research of the National Bureau of Standards 88, 301 (1983)

[7] N. F. Ramsey, Rev. Mod. Phys. 62, 541 (1990)

[8] S. R. Jefferts, T. P. Heavner and E. A. Donley, Jpn. J. Appl. Phys. 43, 2803 (2004)

[9] H. S. Lee et al., J. Korean Phys. Soc. 45, 256 (2004)

[10] J. Vanier and C. Audoin, Metrologia 42, S31 (2005)

[11] J. R. Zacharias, Phys. Rev. 94, 751 (1954)

[12] H. J. Metcalf and P. van der Straten, Laser-Cooling and Trapping (Springer, New York,1999)

[13] W. D. Phillips, Rev. Mod. Phys. 70, 721 (1998)

[14] http://www.nist.gov/pml/div688/grp50/primary-frequency-standards.cfm (4. 5. 2014)

[15] J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards Volume 1(IOP Publishing Ltd, Philadelphia, 1989)

[16] L. L. Lewis, F. L. Walls and D. J. Glaze, J. Phys. Colloq. 42, 241 (1981)

[17] J. H. Shirley, W. D. Lee and R. E.Drullinger, Metrologia 38, 427 (2001)

[18] T. E. Parker, Metrologia 47, 1 (2010)

[19] B. J. Bloom et al., Nature 506, 71 (2014)

[20] http://www.microsemi.com/products/timing-synchronization-systems/embedded-timing-solutions/components/sa-45s-chip-scale-atomic-clock#overview (11. 5. 2014)

[21] C. W. Chou et al., Phys. Rev. Lett. 104, 070802 (2010)

[22] C. W. Chou et al., Science 329, 1630 (2010)

[23] J. D. Prestage, R. L. Tjoelker and L. Maleki, Phys. Rev. Lett. 74, 3511 (1995)

[24] T. Rosenband et al., Science 319, 1808 (2008)

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